Krige, who was group finance director at John Laing for nearly four years, will start in October.
She replaces Justin Read, who resigned with effect from the end of August to take up a new role elsewhere.
Steve Corcoran, Speedy chief executive, said: “On behalf of the company, I am delighted to welcome Lynn to the board.
“Her experience will be most valuable and we look forward to her input as the company continues its growth.
“I would like to thank Justin for all of his hard work and the contribution that he has made to Speedy. I wish him success in his new role.”